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Identification of the Dynamic Parameters of the

Orthoglide

Sylvain Guegan, Wisama Khalil, Philippe Lemoine

To cite this version:

Sylvain Guegan, Wisama Khalil, Philippe Lemoine. Identification of the Dynamic Parameters of the

Orthoglide. IEEE International Conference on Robotics and Automation - ICRA 2003, Sep 2003,

Taipei, Taiwan. pp. 3272-3277, �10.1109/ROBOT.2003.1242095�. �hal-00362674�

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Identification of the Dynamic Parameters of the Orthoglide

Sylvain GUEGAN, Wisama KHALIL, Philippe LEMOINE

Ecole Centrale de Nantes

IRCCyN U.M.R. C.N.R.S. 6597

1 Rue de la Noë, BP 92101, 44321 Nantes Cedex 03, FRANCE

wisama.khalil@irccyn.ec-nantes.fr

Abstract

This paper presents the experimental identification of the dynamic parameters of the Orthoglide [1], a 3-DOF parallel. The dynamic identification model is based on the inverse dynamic model, which is linear in the parameters. The model is computed in a closed form in terms of the Cartesian dynamic model elements of the legs and of the Newton-Euler equation of the platform. The base inertial parameters of the robot, which constitute the identifiable parameters, are given.

1 Introduction

The inverse dynamic model is important for high performance control algorithms, and the forward dynamic model is required for their simulation. For these two applications the numerical values of the dynamic parameters (inertial and friction) must be known. The determination of the base inertial parameters, which represent the only identifiable parameters [2], is treated in this paper by a numerical method [3]. This method is based on the QR decomposition of the observation matrix of the dynamic identification model of the robot. The experimental identification of the dynamic parameters is based on the use of a dynamic model linear in the parameters. This model permits to use the least squares solution to solve the estimation problem [4].

2 Kinematic modeling of the Orthoglide

The Orthoglide has three PRPaR identical legs (where P, R and Pa stand for Prismatic, Revolute and Parallelogram joint, respectively). Each leg is composed of six passive revolute joints and 1 active prismatic joint, (fig. 1). We define frame F0 fixed with the base and frame

FP fixed with the mobile platform (fig. 2). Their origins

are A1 and P respectively. Their axes (x0, y0, z0) and (xP,

yP, zP) are parallel. The base frames of the legs are

defined by the frames FA1, FA2 and FA3 (fig. 2), whose

origins are A1, A2 and A3 respectively. The zAi axes are

along the prismatic joint axes. The Khalil and Kleinfinger notations [5], are used to describe the geometry of the system (fig. 3).

Fig. 1: Orthoglide kinematic architecture.

A1 A2 z0, zA1 y0, yA1 x0 xA1 A3 P zP yP xP F0 FP K zA2 xA2 xA3 yA3 zA3 yA2

Fig. 2: Base frame, platform frame and leg frames.

The following notations are used:

L (3×1) vector of the motorized joint variables:

L

=

[

q

11

q

12

q

13

]

T;

0 p

V (3×1) vector of the linear velocity of the origin of the platform.

The derivative of L and 0Vp with respect to the time are denoted L! and 0V!p respectively.

This work has been supported by the project MAX of the program ROBEA of the department STIC of the French CNRS.

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X4i' R2i R5i X5i X6i D9i D6i Z8i Z9i Z6i Z5i q8i q5i q4i X2i X3i X4i X2i' X7i X8i X9i Z4i Z3i D 8i b7i q7i Bi Ci P Z7i Z2i X1i Z0i Z1i X0i Ai D4i q2i q3i q1i Leg i Link 1 Link 2 Link 3 Link 4 Link 5 Platform

Fig. 3: Link frames of one leg.

The following kinematic models are presented in [6]: i) The inverse kinematic model of the robot:

L! = 0Jp1 0Vp (1)

Where 0J is the inverse Jacobian matrix of the-1P Orthoglide, which is always regular in the working space. ii) The inverse kinematic model of a leg i:

q!i = 0J-1 0i Vp (2)

q!i =

[

q!1i q!2i q!3i

]

T (3)

Where 0J is the inverse Jacobian matrix of the leg i.-1i The velocities of the other joints of each leg can be obtained in terms of q! (see appendix).i

iii) The second order inverse kinematic model of the leg: !!q = Ji 0 -1i

(

0V - J q!p 0!i!i

)

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3 Inverse dynamic model

The inverse dynamic model gives the motorized forces, ΓΓΓΓrobot, in terms of the desired trajectory of the

mobile platform 0 0 0

p p p

P , V , V! . The dynamic model is computed in two steps. First we calculate, the reaction forces of the platform on the legs at point P, which is denoted by fi, then the Newton-Euler equation of the platform is applied to obtain the motor forces [6].

The general form of the inverse dynamic model of a leg i, is written as (see appendix):

ΓΓΓΓi =H q ,q qi

(

i ! !!i, i

)

+0J fT 0i i (5)

Where:

Hi is the inverse dynamic model of leg i, when its terminal point is free.

ΓΓΓΓi is composed of the independent torques/forces of the joints of the leg i, where Γ1i andΓ2i are zero:

ΓΓΓΓi = Γ

[

1i Γ2i Γ3i

] [

T = Γ1i 0 0

]

T (6) Using equation (5) the forces fi can be written as: 0fi = −H q ,q qxi

(

i ! !!i, i

)

+0J-Ti ΓΓΓΓi (7)

Where:

HXi

(

q ,q qi ! !!i, i

)

=0 -TJ H q ,q qi i

(

i ! !!i, i

)

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Hxi is the inverse dynamic model with respect to the position Cartesian space at point P (fig.3) [7][8]. We show that [6]:

(

)

[ ] , , = − ! !! + 0 0 -T i xi i i i p : i i f H q ,q q J ΓΓΓΓ (9) Where [ ] , 0 -T p : i

J represents the ith column of the inverse transpose Jacobian matrix of the robot.

The Newton-Euler equation of the platform is written as (no rotation): = Mp−Mp 0 0 0 P P F V! g (10) With:

0g Acceleration of gravity, referred to frame F

0:

0g=

[

0 g 0

]

T, g=9.81m.s−2 Mp Mass of the platform;

0F

p Total external forces on the platform.

From equations (9) and (10), the dynamic model is given by:

(

)

3 i 1 , =   = + 

! !!  0 T 0 robot Jp Fp Hxi q ,q qi i i ΓΓΓΓ (11)

Different methods can be used to calculate

(

i i i

)

i q ,q q

H ! ,!! [9][10][11]. To reduce the computational

cost, the customized Newton-Euler method, which is linear in the dynamic parameters is used [12].

4 Dynamic identification model

The dynamic model of each leg i can be represented as a linear function of the inertial and friction parameters of the leg Ki. Thus the equation (11) can be written as:

Γrobot = Drobot Krobot (12)

Krobotis the vector of the standard dynamic parameters of the Orthoglide:

Krobot = MPK K KT1 2T T3T (13)

Drobot = D D D DP 1 2 3 (14)

Ki is the vector of the standard dynamic parameters of the leg i, such that:

Ki = Ma1i Fs1i Fv1i χχχχiTT (15) χχχχi =χχχχ1iT " χχχχ5iTT (16) Where:

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- Ma1i is the inertia of the rotor of motor i referred to the

joint side;

- Fv1i is the viscous friction parameter;

- Fs1i is the coulomb friction parameter;

-χχχχiis the vector of the inertial parameters of link i. The standard inertial parameters of the link j (j = 1 to 5, fig. 3) of the leg i are collected in the (10×1) vector:

T ji ji ji ji ji ji ji ji ji ji XX XY XZ YY YZ ZZ MX MY MZ M   =   ji χχχχ (17) Where:

- XXji, …, ZZji are the elements of the inertia matrix;

- MXji, MYji, MZji define the first moments of link ji;

- Mji is the mass of link ji.

Thus, Krobot is a (160×1) vector and Drobot is a (3×160) matrix.

4.1 Base dynamic parameters of the robot

The base dynamic parameters represent the minimum number of parameters from which the dynamic model can be calculated. The dynamic model complexity is reduced when computed by the base dynamic parameters. Besides, they constitute the only identifiable parameters [3]. They can be obtained from the standard parameters, by eliminating the dynamic parameters that have no effect on the dynamic model and by grouping some others.

To determine them, we use a numerical method, which is based on the QR decomposition [4]. First we determine the base parameters of each leg, then we determine the effect of connecting the platform.

There are 14 base parameters for legs 1 and 2. They are given by (i = 1, 2): Ma1Ri, Fv1i, Fs1i, ZZ2Ri, MX2i, MY2Ri,

XX3Ri, XY3Ri, XZ3Ri, YZ3Ri, ZZ3Ri, MX3Ri, MY3Ri, MX4i. Since,

the prismatic joint of leg 3 is along gravity, there are 15 base parameters for leg 3, the grouped inertia Ma1R3 does

not eliminate M1R3 (Whose effect on the force of motor 3

will be constant and equal to –g.M1R3). The grouped

relations are (the index R indicates that some parameters are grouped with that one):

Ma1Ri =Ma1i+M1i+M2i+M3i+M4i+M7i ZZ2Ri =ZZ2i+YY3i+YY4i+D4i2 M4i+YY7i MY2Ri =MY2i+MZ3i+MZ4i+MZ7i 2 3Ri 3i 3i 7i 7i 4i 4i XX =XX −YY +XX −YY −D M XY3Ri=XY3i+XY7i XZ3Ri=XZ3i−D MZ4i 4i+XZ7i (18) YZ3Ri=YZ3i+YZ7i 2 3Ri 3i 7i 4i 4i ZZ =ZZ +ZZ +D M MX3Ri =MX3i+MX7i+D4i M4i MY3Ri =MY3i+MY7i M1R 3=M13+M23+M33+M43+M73

D4i is the distance between the axes of q3i and q4i (fig.3).

Taking into account the three legs and the platform, we obtain that the parameters MX3Ri (i = 1, 2, 3) are grouped

with the mass of the platform and with the parameters: M1Ri3, ZZ2Ri, XX3Ri and ZZ3Ri:

RP P 3R1 3R 2 3R 3 41 42 43 1 1 1 M M MX MX MX D D D = + + +

1Ri 1Ri 3Ri

4i

1

Ma Ma MX

D

= −

ZZ2Ri =ZZ2Ri−D MX4i 3Ri (19)

XX3Ri =XX3Ri+D4iMX3Ri ZZ3Ri =ZZ3Ri−D MX4i 3Ri

1R 3 1R 3 3R 3 43 1 M M MX D = −

To understand the physical meaning of these grouped parameters, let us consider that the center of mass of links 3i and 7i is in the middle of O3iO4i and O7iO8i respectively.

Thus: 4i 3i 3i D MX M 2 = , 4i 7i 7i D MX M 2 = (20)

Using equations (18) and (20) into (19), we obtain: 3 RP P 4i 3i 7i i 1 1 1 M M M M M 2 2 =   = + + +  

Ma1Ri Ma1i M1i M2i 1M3i 1M7i 2 2 = + + + + (21) M1R 3 M13 M23 1M33 1M73 2 2 = + + +

From equation (21), we show that:

- The masses M4i are grouped entirely with the platform;

- The masses M3i and M7i are divided by two: one half is

grouped with the platform and the other with Ma1Ri and

also with M1R3 when i = 3.

The base dynamic parameters of the Orthoglide are given in table 1, on which we added an offset on the motor forces. The masse M1R3 will be grouped with Off3.

Thus the total number of parameters is 43.

Table 1: Base inertial parameters of the Orthoglide

MRP

Ma1Ri Offi Fv1i Fs1i ZZ2Ri MX2i MY2Ri

XX3Ri XY3Ri XZ3Ri YZ3Ri ZZ3Ri MY3Ri MX4i

So, KBrob, which contains the base dynamic parameters of the robot, is written as:

M

RP

T

T T T

Brob B1 B2 B3

K

=

K

K

K

(22)

Where KBi is the vector of the base dynamic parameters of the leg i. The corresponding DBrob matrix can be written as:

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4.2 Computation of DBrob

The vector Hi of the leg i can be written as:

(

)

0 0     =       mi i i i i Li Bi D H q ,q ,q! !! D K (24)

Where the elements of Dmi correspond to the base parameters corresponding to the motorized joint (Ma1Ri,

Fv1i, Fs1i and Offi for i = 1 to 3):

1Ri 1i 1i i Ma Fv Fs Off D D D D   =   mi D (25) With: 1Ri Ma 1i D =!!q , 1i Fv 1i D =q! , 1i Fs 1i D =sign(q )! and i Off D =1. The columns of DLi correspond to the other base parameters.

Then, from (11), and (24), we deduce that:

D = J

P 0 Tp

(

0

V

!

P

0

g

)

(26) 0 0           m1 0 T 0 -T B1 p 1 L1 D D = J J D (27) 0 0           0 T 0 -T B2 m2 p 2 L2 D = D J J D (28) 0 0           0 T 0 -T B3 p 3 L3 m3 D = J J D D (29)

4.3 Exploitation of the similarity of the legs

The complexity of the dynamic identification model could be reduced by making use of the similarity of the legs. Thus, the base parameters ZZ2Ri, MX2i, MY2Ri,

XX3Ri, XY3Ri, XZ3Ri, YZ3Ri, ZZ3Ri, MY3Ri, MX4i of the

three legs could be grouped together. Thus the matrix

Brob D becomes: 0 0 0 0 0 0           m1 sym P m2 LS m3 D D = D D D D (30) 3 i 1=    

0 T 0 -T LS p i Li D = J J D sym

K is a (23×1) matrix containing the base inertial parameters of the Orthoglide:

= sym K (31) T RP 1R1 11 11 1 1R3 13 13 3 2R 4 M Ma Fv Fs Off Ma Fv Fs Off ZZ MX    " "  With: 2R 2R1 2R 2 2R 3 4 41 42 43 ZZ =ZZ =ZZ =ZZ ,", MX =MX =MX =MX

4.4 Identification of the base dynamic parameters

The identification has been carried out using least squares techniques on the dynamic model as described by Gautier in [13],[14]. To identify the base dynamic parameters some trajectories are sampled at different times. The matrix Dsym is calculated for each sample and all of them are collected in the matrix W, to obtain the following overdetermined linear system of equations:

Y = W Ksym+ρρρρ (32)

Where:

ρρρρ is the modeling error;

W is the (3r×c) observation matrix and Y is the (3r×1) matrix corresponding to the joint forces for all the samples, with c = 23 and r represents the number of the samples.

The solution of the linear system of equations (32) gives the estimation of the base dynamic parameters.

5 Experimental results

The identification method has been experimentally carried out on the Orthoglide prototype of the IRCCyN. The motors are AC servomotors. The control system is based on a DSPACE 1103 digital signal-processing card. The sampling period is 2.5 ms.

5.1 Planning of the identification trajectory

To identify the base dynamic parameters the choice of the robot trajectory is very important in order to excite the different parameters [15]. The condition number of W has been used to select the best trajectory. This number measures the sensitivity of the solution with respect to the noise in the data. The Orthoglide motorized joints could be derived independently. So trajectories have been generated between random joint positions. By simulation we selected 10 random trajectories giving a good condition number. The trajectories are then executed on the real system and sampled with a period which is equal to 2.5 ms. The matrix Dsym is calculated for each sample and all of them are collected in the observation matrix W.

5.2 Estimation of the observation matrix

The computation of the Dsym matrix needs the estimation of the joint positions, velocities and accelerations. The joint positions are measured thanks to the digital encoder.

The joint positions have been filtered with a 4th-order low-pass Butterworth filter in both the forward and reverse directions to avoid phase distortion. The corresponding cut-off frequency is 100 Hz. The joint velocities and accelerations are calculated using numerical derivation based on a central difference

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algorithm. In fact, low-pass filtering associated with a difference algorithm provides a pass-band filter to estimate derivatives at low frequency, which avoid derivating high frequency noise [13].

In order to eliminate high frequency torque noises and ripples from Y, the columns of W and the vector Y are filtered in a process called parallel filtering, using the function "decimate" of order 5 from Matlab [13].

5.3 Estimation of the dynamic parameters

The least squares solution has been applied to relation (32) in order to estimate the dynamic parameters:

ˆKsym = W Y+ (33)

Where W is the pseudo-inverse of W.+

The standard deviations are estimated considering the matrix W to be deterministic one, and ρρρρ to be a zero mean additive independent noise, with standard deviation

ρρρρ

σσσσ . The variance-covariance matrix of the estimation error and standard deviations can be calculated by [13]:

ˆ

(

)

1 − = 2 T K C σσσσρρρρ W W (34) ˆ ˆ Ki CKi σ = (35)

Where σσσσρρρρ is obtained by the expression: ˆ ˆ 3 r c = 2 2 Y - W K -ρρρρ σσσσ (36)

The relative standard deviation is given by:

Kiˆ ˆ Kri i 100 ˆ K σ σ = (37)

Table 2 gives the estimated base dynamic parameters and their relative standard deviations:

Table 2: Estimation of the base dynamic parameters

ˆ K σKriˆ Kˆ σKriˆ MRP 3.2555 3.4208 Off3 -2.6104 -23.524 Ma1R1 8.9738 0.4971 ZZ2R -2.8560 -31.5797 Fv11 84.7633 0.5559 MX2 -4.6706 -31.0705 Fs11 54.35 0.4534 MY2R -0.0200 -29.4402 Ma1R2 8.843 0.4904 XX3R 1.5458 30.7643 Fv12 67.1047 0.7118 XY3R 0.0127 23.2455 Fs12 80.5591 0.3105 XZ3R 0.0378 14.5685 Ma13 8.7612 0.5125 YZ3R -0.0088 -22.8285 Fv13 83.7167 0.5669 ZZ3R -0.1356 -10.103 Fs13 54.9855 0.4426 MY3R 0.0120 43.8424 Off1 -3.0074 -7.8658 MX4R 4.5554 31.827 Off2 -1.8386 -11.989 Size of W: 65814 x 23 Condition number of W: 4805.57

A parameter with σ ≥Kriˆ 15 is considered to be not

good identified [15], it may have a little effect on the model and cannot be identified with acceptable accuracy

on the actual trajectory. Taking into account that the parallelograms links are light and symmetric, and considering the grouped relations, we find that ZZ2R,

MX2, …, MX4 are close to zero and can be neglected.

Thus, we repeat the identification process with the following essential parameters (table 3):

Table 3: Estimation of the essential base dynamic parameters ˆ K σKriˆ Kˆ σKriˆ MRP 1.5993 1.2601 Ma13 8.6790 0.3487 Ma1R1 8.7002 0.3128 Fv13 83.7047 0.5700 Fv11 84.6695 0.5594 Fs13 54.9723 0.4451 Fs11 54.4006 0.4554 Off1 -2.5644 -4.8819 Ma1R2 8.4992 0.3210 Off2 -0.4616 -27.1756 Fv12 66.888 0.7177 Off3 -10.1975 2.3957 Fs12 80.7283 0.3114 Size of W: 65814 x 13 Condition number of W: 77.69

We note that the identified values are in accordance with respect to our knowledge of the system. The parameters MRP is near the a priori value, which is 1.5 Kg.

The estimation errors of each parameter σXriˆ are

acceptable (except Off2, which could be neglected). The

condition number of the observation matrix is very good. We note that the effect of the rotor inertia, viscous friction and coulomb friction of the motorized joints are important compared to the effect of the base parameters of the parallelogram links. The dynamic model corresponding to the parameters of table 3 is easy to compute on line for control purpose.

5.4 Validation of the results

Two main validation procedures have been carried out: i) The comparison of the estimated torques with respect to the measured torques on the trajectory that have been used in the identification, and with some other trajectories that have been not used in the identification.

ii) The addition of a payload on the platform to observe the evolution of the mass parameter of the platform MRP.

All these tests show very good results.

6 Conclusion

This paper presents the identification of the dynamic parameters of the Orthoglide, a 3-DOF parallel robot that moves in the Cartesian space with fixed orientation. The dynamic identification model is based on the inverse dynamic model, which is linear in the parameters. The model is computed in terms of the Cartesian dynamic model elements of the legs and of the Newton-Euler equation of the platform. The base inertial parameters of the robot, which constitute the minimum number of inertial parameters, are determined using a numerical

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method using the QR decomposition. We proposed to make use of the similarity of the legs in order to reduce the number of parameters and to improve the condition number of the observation matrix. Experimental results are presented, and the validation is very good. Future work will concern the use of the identified model to control the robot using a dynamic control law.

References

[1] Wenger, P., and Chablat D., "Kinematic analysis of a new parallel machine tool : the Orthoglide", Advances in Robot Kinematic, Kluwer Academic Publishers, 2000, pp.305-314.

[2] Mayeda H., Yoshida K., Osuka K., "Base parameters of manipulator dynamic models", IEEE Trans. on Robotics and Automation, Vol. RA-6(3), 1990, pp. 312-321.

[3] Gautier, M., "Numerical calculation of the base inertial parameters", IEEE Int. Conf. on Robotics and Automation, Cincinnati, U.S., 1990, pp.1020-1025 [4] Gautier, M., and Khalil, W., "Direct calculation of minimum set of inertial parameters of serial robots", IEEE Trans. on Robotics and Automation, Vol6(3), 1990, pp.368-373.

[5] Khalil, W., and Kleinfinger, J.-F., "A new geometric notation for open and closed-loop robots", Proc. IEEE Conf. on Robotics and Automation, San Francisco, US, 1986, pp.1174-1180.

[6] Guegan, S., and Khalil, W., "Dynamic modeling of the Orthoglide", Advances in Robot Kinematic, Kluwer Academic Publishers, Caldes de Malavella, Spain, 2002, pp.387-396.

[7] Khatib, O., "A unified approach for motion and force control of robot manipulators: the operational space formulation", IEEE J. of Robotics and Automation, Vol. RA-3(1), 1987, pp.43-53.

[8] Lilly, K.W., and Orin D.E., "Efficient O(N) computation of the operational space inertia matrix", Proc. IEEE Int. Conf. on Robotics and Automation, Cincinnati, US, 1990, pp.1014-1019.

[9] Khalil, W., and Dombre, E., "Modeling, identification and control of robots", Hermes Penton, 2002, London-Paris.

[10] Luh, J.Y.S, Walker, M.W., and Paul, R.C.P., "On-line computational scheme for mechanical manipulators",

Proc. 2nd IFAC/IFIP Symp. on Information Control

Problems in Manufacturing Technology, Stuttgart, 1979, pp.165-172.

[11] Craig J.J., "Introduction to robotics: mechanics and control", Addison Wesley Publishing Company, Reading, 1986.

[12] Khalil, W., and Kleinfinger J.-F., "Minimum operations and minimum parameters of the dynamic model of tree structure robots", IEEE J. of Robotics and Automation, Vol. RA-3(6), 1987, pp.517-526.

[13] Gautier M., "A comparison of filtered models for dynamic identification of robots", Proc. IEEE 35th Conf. on Decision and Control, Kobe, Japon, déc. 1996, pp. 875-880.

[14] Gautier M., Khalil W., Restrepo P.P., "Identification of the dynamic parameters of a closed loop robot", Proc. IEEE Int. Conf. on Robotics and Automation, Nagoya, mai 1995, pp. 3045-3050.

[15] Gautier M., Khalil W., "Exciting trajectories for inertial parameters identification", The Int. J. of Robotics Research, Vol. 11(4), 1992, pp. 362-375.

Appendix : Dynamic model of a leg

Each leg has a planar parallelogram closed loop. The inverse dynamic model of the equivalent tree structure is obtained by cutting the revolute joint q8i (i = 1, 2, 3),

figure 3:

ΓΓΓΓtri = Htri

(

q ,q ,qi ! !!i i

)

(38) Leg i is isolated from the platform, so we can consider the variables q1i, q2i and q3i to be independent. In the

complete model of the robot, only q1i is active and the

torques Γ2i and Γ3i are zero.

Let the vector q be composed of the independentai joints and the vector q be composed of the passive jointspi of leg i: =

[

q1i q2i q3i

]

T i a q =

[

q4i q5i q7i

]

T i p q (39)

The constraint equations of the loop are:

q4i q , q3i 5i q2i , q7i q , q3i 8i q3i 2

π

= − = − − = = − (40)

The dynamic model of the leg is obtained from Γtriand the constraint equations by [9]:

(

)

[

]

T 1i 2i 3i = = Γ Γ Γ = i i i i i i T i i tr H q ,q ,q G ΓΓΓΓ Γ Γ Γ Γ Γ Γ Γ Γ ! !! (41) With: =∂ ∂ i i tr i a q G q and i  i i T T T tr a p q = q q , thus: 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1     =  −    T i G (42)

The dynamic model of the tree structure is obtained by recursive symbolic Newton-Euler method [12].

Since all the parameters of the 5th and 7th links are grouped with the other links. Thus the matrix G can be reduced to the first 4 columns:

1 0 0 0 0 1 0 0 0 0 1 1     =    −    T i G (43) And: ΓΓΓΓtri = ΓΓΓΓtr1i " ΓΓΓΓtr4iT (44)

Figure

Fig. 1: Orthoglide kinematic architecture.
Fig. 3: Link frames of one leg.
Table 1: Base inertial parameters of the Orthoglide
Table 2: Estimation of the base dynamic parameters

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